A337367 Sum of square end-to-end distance over all self-avoiding n-step walks on a square lattice where no adjacent points are allowed, except those for consecutive steps.

0, 4, 32, 156, 608, 2116, 6816, 20844, 61376, 175628, 491248, 1349172, 3650144, 9751532, 25774672, 67501556, 175375136, 452454276, 1160098576, 2958123556, 7505767840, 18959922796, 47701159264, 119570463980, 298719578688, 743984084700, 1847709517360, 4576818079076, 11309417827072

Offset: 0

Author

Scott R. Shannon, Aug 25 2020

Keywords

• nonn
• walk

Comments

The corresponding number of n-step walks is given in A173380.

Examples

			The allowed 4-step walks with their associated end-to-end square distances are:
.
         + 10
4        |        8              8      8           16
+--+     +     +--+              +      +    X--+---+---+---+
   |     |     |          10     |      |
   +     +     +     +--+--+  +--+      +        +--+ 10      + 10
   |     |     |     |        |         |        |            |
X--+  X--+  X--+  X--+     X--+   X--+--+  X--+--+   X--+--+--+
.
The eight non-straight walks sum to 68, and these can be walked in eight ways on the square lattice. The remaining straight walk can be walking in four ways. Thus a(4) = 68 * 8 + 16 * 4 = 608.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the sequence A173380).

Links

• Sequence Fans Mailing list, discussion of the sequence A173380, November 2010.

Crossrefs

Cf. A173380, A001411.